Extraction of Neutrino Flux from Inclusive Neutrino-Nucleus Reaction Osaka Univ. Tomoya Murata, Toru Sato contents １．Introduction ２．Maximum Entropy Method(MEM) ３．Cross Section of Neutrino Nucleus Reaction ４．’Data’ of muon energy distribution ５．Results ６．Summary １．Introduction Neutrino Oscillation 𝜈𝜇 The neutrino flavor changes during propagating 𝜈𝜏 Flavor eigen states are expressed by linear combination of mass eigen states −𝑖𝛿 𝑐12 𝑐13 𝜈𝑒 𝜈𝜇 = −𝑠12 𝑐23 − 𝑐12 𝑠23 𝑠13 𝑒 −𝑖𝛿 𝜈𝜏 𝑠12 𝑠23 − 𝑐12 𝑐23 𝑠13 𝑒 𝑖𝛿 Flavor eigen state 𝑠12 𝑐13 𝑐12 𝑐23 − 𝑠12 𝑠23 𝑠13 𝑒 𝑖𝛿 −𝑐12 𝑠23 − 𝑠12 𝑐23 𝑠13 𝑒 𝑖𝛿 𝑠13 𝑒 𝑠23 𝑐13 𝑐23 𝑐13 𝜈1 𝜈2 𝜈3 Mass eigen state PDG 2013 −3 2 Δm223 = 2.35+0.12 −0.09 × 10 𝑒𝑉 2 −5 2 Δ𝑚12 = 7.58+0.22 −0.26 × 10 𝑒𝑉 𝑚3 > 𝑚1 , 𝑚2 > 𝑚3 ? ? CP symmetry breaking in lepton sector Mass hierarchy Determination of Neutrino Mixing Parameter Ex.) Oscillation probability in two flavor model We must know the absolute value and the energy dependence of Neutrino Flux Conventional method to determine the flux Quasi Elastic reaction 𝜇− 𝜈𝜇 reconstruct the neutrino energy by energy momentum conservation law 𝜇− 𝜈𝜇 n p Flux averaged cross section We can observe only the flux averaged cross section. 𝑑2 𝜎 𝑑Ω𝜇 𝑑𝐸𝜇 Neutrino Flux T2K PRD87,012001(2013) Neutrino-nucleus charged current reaction cross section 𝜃𝜇 = 30∘ 𝐸𝜈 = 600𝑀𝑒𝑉 𝐸𝜈 = 780𝑀𝑒𝑉 𝐸𝜈 = 940𝑀𝑒𝑉 𝐸𝜈 = 1140𝑀𝑒𝑉 Various reaction mechanisms 𝐸𝜈 𝐸𝜇 , 𝜃𝜇 𝐸𝜇 − 𝑚𝜇2 /(2𝑀) = 1 − (𝐸𝜇 − 𝑃𝜇 cos 𝜃𝜇 )/𝑀 Reaction mechanisms at QE region 𝜇 𝑝 QE with FSI violates the above simple relation MEC FSI 𝜋 𝜋 abs. 𝑁 If pion or nucleon cannot be observed, they are misidentified as QE We need to reconstruct the neutrino flux The Objective Without assuming QE, using only information of inclusive cross section, can we reconstruct the neutrino flux? observable Flux averaged distribution of muon energy and scattering angle Inclusive cross section of muon production Neutrino Flux Reconstruct the Neutrino Flux by using Maximum Entropy Method ２．Maximum Entropy Method (MEM) Maximize the probability 𝑃[Φ|𝐺, 𝐼] → Neutrino Flux ・𝑃 ΦΦ(𝐸 𝐺, 𝐼 𝜈:) experimental data 𝐺、prior information of flux 𝐼, the probability of having Φ(E𝜈 ) . ・experimantel data 𝐺 : (𝐺𝑙 , 𝜎𝑙 ) ・・・ energy and scattering angle distribution of muon ・prior information of flux 𝐼 : (𝑚𝑖 )・・・guess of neutrino flux Φ 1 2 𝑃 Φ 𝐺, 𝐼 ∝ exp − 𝜒 − 𝛼𝑆 2 𝜒2 = 𝐺𝑙 −𝐺𝑙 𝑙 𝜎𝑙 2 ,𝑆= 𝑖 Φ𝑖 − 𝑚𝑖 − Φ𝑖 ln find the flux Φ(𝐸𝜈 ) so that minimize this quantity Φ𝑖 𝑚𝑖 the prior information of the flux. (default model) ３．Cross Section of Neutrino Nucleus Reaction 𝜈𝜇 + 12𝐶 → 𝜇− + 𝑋 𝐸𝜈 =2GeV 𝐸𝜈 =1GeV Pion production 𝜇− 𝜋 𝜈𝜇 T.Sato , D.Uno , T.S.H.Lee dynamical 𝑑2 𝜎 𝑑Ω𝜇 𝑑𝐸𝜇 (𝜃𝜇 = 10∘ ) 𝐸𝜈 =3GeV 𝐸𝜈 =4GeV 𝐸𝜈 =5GeV QuasiElastic 𝜇− 𝜈𝜇 B.Szczerbinska , T.Sato , K.Kubodera , T.S.H.Lee PLB649 132 ４．’Data’of muon energy distribution (flux detected at Near Detector of T2K) × (Oscillation probability of two flavor approximation) Δ𝑚2 = 2.5 × 10−3 𝑒𝑉 2 𝐿 = 295𝑘𝑚 sin2 2𝜃23 = 1.00 error 𝜎＝７％ Gaussian distribution Resolution of muon energy 100MeV ５．Result reconstruct the neutrino flux from‘data’ by MEM ME M Reconstructed Neutrino Flux (𝜃𝜇 =10°) ‘data’of muon distribution Reconstructed Neutrino Flux ME M Flux was reconstructed successfully. Dependence of Default Model (𝜃𝜇 =10°) Default Model || The prior information of the flux Original Flux Reconstructed Flux by MEM Default Model Original Flux Without oscillation Constant Dependence of default model is not much. We use the model of without oscillation. Change the error sizes (𝜃𝜇 =10°) ‘data’of muon distribution ME M Reconstruct ed Neutrino Flux < Φ > = 42.3 ± 2.0 < Φ > = 43.5 ± 2.9 ６．Summary We proposed a method of reconstruction the neutrino flux from muon distribution in inclusive neutrino nucleus charged current reaction. We regarded the reconstruction problem as the Inverse problem and use maximum entropy method. ME M reconstruct neutrino flux without assuming CCQE. The neutrino flux was successfully reconstructed. Future Works • Construction of the accurate model of neutrino nucleus reaction • Study 𝜈𝑒 appearance reaction.(𝛿𝐶𝑃 ) • Examine the method for the deep inelastic scattering region. Thank You .
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